Question:

<p><strong>Step 1: Analyze the condition z + z̄ = 2|z - 1|</strong></p><p>Let z = x + iy. Then z + z̄ = 2x and |z - 1| = √[(x-1)² + y²]</p><p>So: 2x = 2√[(x-1)² + y²]</p><p>x = √[(x-1)² + y²]</p><p>Squaring: x² = (x-1)² + y²</p><p>x² = x² - 2x + 1 + y²</p><p>2x = 1 + y²</p><p>y² = 2x - 1</p><p>This is a parabola with vertex at (1/2, 0).</p><p><strong>Step 2: Interpret arg(z₁ - z₂) = π/4</strong></p><p>The argument of z₁ - z₂ is π/4, meaning the direction from z₂ to z₁ makes an angle of π/4 with the positive real axis.</p><p>If z₁ - z₂ = r·e^(iπ/4) = r(cos(π/4) + i·sin(π/4)) = r(1/√2 + i/√2) for some r > 0</p><p>Then: Im(z₁ - z₂) = r/√2 and Re(z₁ - z₂) = r/√2</p><p>So Im(z₁ - z₂) = Re(z₁ - z₂)</p><p><strong>Step 3: Use both conditions simultaneously</strong></p><p>Both z₁ and z₂ lie on the parabola y² = 2x - 1.</p><p>Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where y₁² = 2x₁ - 1 and y₂² = 2x₂ - 1.</p><p>From arg(z₁ - z₂) = π/4:</p><p>y₁ - y₂ = x₁ - x₂</p><p><strong>Step 4: Solve for the points</strong></p><p>y₁² - y₂² = 2(x₁ - x₂) = 2(y₁ - y₂)</p><p>(y₁ - y₂)(y₁ + y₂) = 2(y₁ - y₂)</p><p>Since z₁ ≠ z₂, we have y₁ - y₂ ≠ 0, so:</p><p>y₁ + y₂ = 2</p><p><strong>Step 5: Verify with a geometric argument</strong></p><p>The parabola y² = 2x - 1 is symmetric about the x-axis. For the argument condition to be satisfied with two distinct points on the parabola, by symmetry and the constraint, the sum of imaginary parts equals 2.</p><p><strong>∴ Answer: Im(z₁ + z₂) = 2</strong></p>